Inthis example, winter (DJF) meanprecipitation from anumber of rain gauges on the Iberian Peninsula is related tothe air-pressure field over the North Atlantic. CCA was usedto obtain a pair of canonical correlation pattern estimates(Figure 14.4), and corresponding time series of canonicalvariate

estimates. These strongly correlated modes ofvariation (the estimated canonical correlation is 0.75)represent about 65% and 40% of the total variability ofseasonal mean SLP and Iberian Peninsula precipitationrespectively.

The analysis described above wasperformed withthe 1950-80 segment of a data setthat extendsback to 1901. Sincethe 1901-49segment isindependent of that used to 'train'themodel,it can be used to validate the model

Figure14.5 shows both the specified andobserved wintermean rainfall averaged over allIberian stationsfor thisperiod. The overall upwardtrend andthe low-frequencyvariations inobserved precipitationare well reproduced bytheindirect methodindicating the usefulness of thetechnique aswell as the reality of both the trendand thevariations in the Iberian winter precipitation.

14.3.4North Atlantic SLP andIberian Rainfall:Downscaling of GCMoutput

This regression approach has an interestingapplication in climate changestudies. GCMsarewidely used to assess theimpact thatincreasingconcentrations of greenhousegases mighthave onthe climate system. But, becauseof theirresolution,GCMs do not represent thedetails ofregional climatechange well. Theminimum scalethat a GCM is able toresolve is thedistance betweentwoneighboringgridpoints whereasthe skillfulscale is generally acceptedto be fouror moregrid lengths. The minimum scale inmost climatemodels in the mid 1990s is of theorderof250-500 km so that theskillfulscale is atleast1000-2000 km.

Thus the scales at which GCMsproduce usefulinformation does not match the scaleat whichmanyusers, such as hydrologists,require information.Statistical downscalingis a possiblesolution to thisdilemma. Theidea isto build a statistical model fromhistorical observationsthat relates large-scaleinformation thatcan be well simulated by GCMs tothe desiredregional scale information that can not be

simulated. These models are then applied tothelarge-scale model output.

The following steps must betaken.

1. Identifya regional climate variable R of

Interest

2. Finda climate variableL

that:

•controlsR in the sense that there isastatisticalrelationship betweenR

andL

ofthe form

R = G(L,a) + ein whichG(L,a)

represents asubstantial fractionof the total variance of R.Vectora

contains parametersthat can be usedtoadjust the fit

•isreliably simulated in a climate model.

3. Usehistorical realizationsof(R, L)to estimatea.

4. Validatethe fitted model onindependenthistorical data

5. Applythe validated model to GCMsimulatedrealizationsofL.

Thisareexactlythe steps taken in the Atlantic SLPand Iberian precipitation analysis.

with the model'sgrid pointresponse. The latter suggests that therewill beamarked decrease in precipitation overmost ofthePeninsula whereas the downscaledresponse isweakly positive. The downscaledresponse isphysically more reasonable than thedirectresponseof the model.

of the data pointsalong the dotted lines tobe higherthan thatalong the dashed lines (where the data points a,b,c andd in the x-space are ordered as b, a, d andc in the y-space),the dottedlines are chosen asthefirstCCA mode.

Maximum covariance analysis (MCA),looksformodesof maximumcovariance instead ofmaximum correlation,and would select thedashed lines over the dotted lines since thelength of the linesdo countin the covariancebut not in thecorrelation.

It can be shown that the MCA problem can bederived from CCA by pre-filtering the data usingPrincipal Components (EOFs) of the data.

But a more straightforward derivation is obtainedusing a different normalization before using themethod of Lagrange multipliers.